Bruger:Inc/Sandkasse 2

Inden for matematik og signalbehandling er Hilberttransformationen en lineær operator, der tager funktionen $u(t)$ og producerer $H(u)(t)$. Operatoren er givet ved en foldning med funktionen $1/(\pi t)$:

$H(u)(t)={\frac {1}{\pi }}\int _{-\infty }^{\infty }{\frac {u(\tau )}{t-\tau }}\,d\tau ,$

the improper integral being understood in the principal value sense. The Hilbert transform has a particularly simple representation in the frequency domain: it imparts a phase shift of 90° to every Fourier component of a function. For example, the Hilbert transform of $\cos(\omega t)$, where ω > 0, is $\cos(\omega t-\pi /2)$.

The Hilbert transform is important in signal processing, where it derives the analytic representation of a real-valued signal u(t). Specifically, the Hilbert transform of u is its harmonic conjugate v, a function of the real variable t such that the complex-valued function u + iv admits an extension to the complex upper half-plane satisfying the Cauchy–Riemann equations. The Hilbert transform was first introduced by David Hilbert in this setting, to solve a special case of the Riemann–Hilbert problem for analytic functions.

Kategori:Funktioner